Download Electromagnetic Induction

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Electromagnetic
Induction
induction (in-DUK-s/mnJ II.: the process by which
an electricor IIUIRnetic
field is created ill ulwthcr
object, usually a conductor.
iNDUCED CURRENTS
OII.IECTIVES
. Study the distinction between
an applied emf and an induced
emf.
. Discuss Faraday's laws.
.
Describe the factors that affect
an induced emf.
. Describe the generator
principle.
. Study instantaneous current and
voltage.
. Describe the motor principle.
. Study mutual inductance and
self-inductance.
.
Describe the transformer.
20.1 Discovery of lnduced Current
In Section 19.8 we
discussed Oersted's discovery of the link between magnetism and electricity. Soon after Oersted's work, other
scientists attempted to find out 'whether an electric current
could be produced by the action of a magnetic field. In
1831 Michael Faraday discovered that 1111 emf is set up inn
closed electric circuit located in a magnetic field wherlever tile
totallllagnetic flux linking the circuit is changing. The American physicist Joseph Henry (1797-1878) made a similar
discovery at about the same time. This phenomenon
is
calJed electromagnetic induction. The emf is called an induced emf, and the resulting current in the closed conduding loop is called an induced current.
The production and distribution of ample electric energy are essential functions of a modern technological
economy. The invention of electric generators and transformers has made it possible to provide these essential
services. The discoveries of Faraday and Henry represent
the initial step in the development of the broad knowledge
base in electromagnetic induction that makes such inventions possible.
20.2 Faraday's
Induction
Experiments
We shall examine some of Faraday's experiments in order to understand their significance. Suppose a sensitive galvanometer
is connected in a closed conducting loop as shown in Figure 20-1. A segment of the conductor is poised in the field
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ELECTROMAGNETIC INDUCTION
491
flux of a strong magnet. As the conductor in Figure 20l(A) is moved down between
the poles of the magnet,
there is a momentary deflection of the galvanometer needle, indicating an induced current. The needle shows no
deflection when the conductor is stationary in the magnetic flux. This observation suggests that the induced current is related to the motion of the conductor in the nwgnetic
flux.
As the conductor is raised between the poles of the magnet, as in Figure 20-1(8), there is another momentary de- Magnetic lines of flux always link
flection of the galvanometer, but in the opposite direction.
the current loop that sets up the
This suggests that the direction of the induced current in the magnetic field. Review Sectioll
conductor is related to the direction of 1110ti0l1
of the conductor ill 19.9.
ti,e magneticfield. The emf induced in the conductor is of
opposite polarity to that in the first experiment.
Faraday found that he cou]d induce an emf in a conductor either by moving the conductor through a stationary
field or by moving the magnetic field near a stationary
conductor. He observed that the direction of the induced
current in the conducting loop is reversed with a change in
either the direction of motion or the direction of the magnetic field.
Supporting the conducting loop of Figure 20-1 in a fixed
position and lifting the magnet result in a deflection similar to that of Figure 20-1(A). When the magnet is lowered,
Figure 20-1. A current is induced
the galvanometer need1e is momentarily deflected as in in a closed conducting loop when
the magnetic flux linked through
the circuit is changing.
t
,-
(A)
(B)
492
CHAPTER
The conducting
segment
of Figure
of a conducting loop
that includes a galvanometer as
shown in Figure 20-1.
20-2
is part
Figure 20-2. An emf is induced in
a conductor when there is a
change of flux linked by the conductor.
Figure 20-1(B). The relative motion between the conductor
and the magnetic flux is the same whether the conductor
is raised through the stationary field or the field is lowered
past the stationary conductor.
Observe that the relative motion of the conductor in
each of these experiments is perpcndicular to the magnetic
flux. If the conductor in the magnetic field is now moved
parallel to the lines of flux, the galvanometer shows no deflection. A conductor that moves perpendicular
to the
magnetic flux can be construed to "cut through" hnes of
flux and experience a changein flux linkage. Conversely, a
Noinduced emf
(A) Change in flux linkage
Conductor
,Y'
(B) No change
~
in flux linkage
conductor that moves parallel to the magnetic field does
not "cut through" lines of flux and does not experience a
change in flux linkage. See Figure 20-2. These observations suggest that electromagnetic induction results from those
relative motions between conductors and maglletic fields which
are accompanied by changes in magnetic flux linkage.
Suppose the conductor is looped so that several turns
are poised in the magnetic field, as in Figure 20-3. When
the coil is moved down between the poles of the magnet
as before, there is a greater deflection on the galvanometer. By increasing the rate of motion of the coil across the
magnetic flux or by substituting a stronger magnetic field,
greater deflections are produced. In each of these cases the
effect is to increase the number of flux lines "eut" bv turns
of the conductor in a given length of time. We ca'n then
state that the magnitude of the inducedemf, or of the illduced
current in a closed loop, is related to the rate at which the flux
linked by the conductor
Figure 20-3. A greater change in
flux linkage occurs when several
turns of a conductor cut through
the magnetic flux.
20
changes.
20.3 Factors Affecting
lnduced
emf Faraday found
that an emf is inducea in a conductor whenever any
change occurs in the magnetic flux (lines of induction)
linking the conductor.
Very precise experiments
have
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ELECTROMAGNETIC INDUCTION
shown that the emf induced in each turn of a coiled conductor is
proportional to the time rate of change of magnetic flux linking
each turn of the coil.
The total magnetic flux linking the coil is designated by
the Greek letter ~ (phi). If dCPrepresents the change in
magnetic flux linking the coil during the time interval dt,
the emf ~ induced in a single turn of the coil can be expressed as
By introducing a proportionality
constant k, the value of
which depends on the system of units used, the expression can be written as
Where ~ is measured in volts and <Pin webers, the numerical value of k is unity and the equation becomes
Thus, a change in the magnetic flux linking a coil occurring at the rate of 1 weber per second induces an emf of 1
volt in a single tum of the coil.
A coil consists of turns of wire that are, in effect, connected in series. Therefore, the emf induced across the coil
is simply the sum of the emfs induced in the individual
turns. A coil of N turns has N times the emf of the separate
turns. This relationship can be represented by the following equation
The negative sign merely indicates the relative polarity of
the induced voltage. It expresses the fact that the induced
emf is of such polarity as to oppose the change that induced it, a
basic energy conservation principle discussed in detail in
Section 20.6.
To illustrate this concept, suppose a coil of 150 turns
linking the flux of a magnetic field uniformly is moved
perpendicular to the flux and a change in flux linkage of
493
494
CHAPTER
3.0 X 10-5 weber occurs in 0.010 second.
is then
'f,
(Wb
~
The induced
20
emf
- 0.45 V
N'm/A
~
5
5
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Figure 20-4. A force acts on a
moving charge in a magnetic
field.
2004 The Cause of an Induced emf A length of conductor moving in a magnetic field has an emf induced across it
that is proportional to the rate of change of flux linkage.
However, an induced current persists only if the conductor
is a part of a closed circuit. In order to understand
the
cause of an induced emf, we shall make use of several
facts that have already been established.
A length of copper "wire poised in a magnetic field, as
shown in Figure 20-4, contains many free electrons, and
these moving charges constitute an electric current. In Section 19.12 we recognized that a force acts on the movable
coil of a galvanometer in a magnetic field when a current is
in the coil. This force, in effect, acts on the moving charges
themselves. We shall can this force a magnetic force to
distinguish it from the force exerted on free electrons by
the electrostatic field of a stationary charge.
Suppose the copper wire of Figure 20-4, at right angles
to the uniform magnetic field, is pushed downward (into
the page) with a velocity v through the magnetic field of
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ELECTROMAGNETIC
495
INDUCTION
flux density B. As a consequence of the motion of the wire,
the free electrons of the copper conductor may be considered to move perpendicular to the flux with the speed v. A
magnetic force, F, acts on them in a direction perpendicular to both Band v. In response to the magnetic force,
these electrons move toward end a and away from end b.
Because the two ends of the conductor are not connected
in a circuit that would provide a closed path for the induced electron flow, end a acquires a growing negative
charge while a residual positive charge builds up on
end b. Thus a difference of potential is established across
the conductor with a the negative end and b the positive
end. If either the motion of the wire or the magnetic field
is reversed, the direction of F in Figure 20-4 is reversed
and an emf is induced so that end a is positive and end b
is negative.
The accumulations of charges at the ends of the conduc.
tor establish an electric field that increasingly opposes the
movement of electrons through the conductor. The force
of this electric field acting on the electrons from a toward b
soon balances the magnetic force arising from the motion
of the conductor, and the flow of electrons ceases.
The equilibrium potential differenceacross the open conductor
is numerically equal to the induced emf and depends on the
length, l, of wire linking the magnetic flux; the flux density, B, of the field; and the speed, v, with which the conductor is moved through the field.
A changing magnetic field induces
an electric field (Faraday's
law).
The movement of electrons
through the open conductor of
Figure 20-4 comprises a "transient" induced current of extremely short duration.
'I: Hlv
When B is in newtons/ampere
meter (or webers/meter2), 1 Flux density, B, is also called
jg is given in volts.
in meters, and v in meters/second,
magnetic induction.
Assume that the length of the conductor linking the
magnetic field in Figure 20-4 is 0.075 m and the flux den~
sity is 0.040 N/A . m. If the conductor is moved down
through the flux with a velocity of 1.5 mis, the induced
emf is
~
'I: ~ Hlv
~ = 0.040 N/A'm x 0.075 m x 1.5 m/s
'(g = 0.004 5 V or 4.5 m V
In Figure 20-4 a magnetic force equal but opposite to F
will act on the positively charged protons in the copper
nuclei. Since these are in the bound parts of the copper
atoms, they will not move in response to this force. But,
positively charged ions in a liquid or gas would move.
The vectors B, v, and F are all mutually perpendicular as
shown in Figure 20-5. If a charge Q moves with a velocity v
CHAPTER
496
To compare the response of a
moving electric charge to an elec~
tric force and to a magnetic force,
see Figures 16-11 and 20-5.
Verify that if B is expressed in
N/A. m, <t>can be expressed in
N-m/A.
20
B, the force, F,
through a magnetic field of flux density
acting on the charge becomes
QvB
F
~
The force F is in newtons, Q is in coulombs, v is in metersl
second, and B is in newtons/ampere-meter
(or webers/
meter').
n
n
'AI
'BI
Figure 20-5_ The magnetic force
acting on a charge moving in a
magnetic field. (A)The charge is
positive and the force is directed
out of the page. (B) The charge is
negative and the force is directed
into the page.
20.5 The Direction of lnduced Current
Now consider
that a length of conductor is used outside the magnetic
flux to connect ends a and b of the copper wire in Figure
20-4, thus providing a closed-loop path for electron flow.
We shall call the length of wire linked with magnetic flux
the internal circuit and the rest of the conducting path the
external circuit.
When the straight conductor is pushed downward, as in
Figure 20-4, electrons move from the negative end a
through the external circuit to the positive end band
through the internal circuit from end b to end a. Over the
external path from a to b, electrons transform the potential
energy acquired in the internal circuit into kinetic energy
that they expend in the external circuit. In the internal
path, the electric force acting on the electrons from a to b
is reduced below its equilibrium value because of the partial depletion of accumulated charges at a and b. A net
magnetic force pumps electrons from b to a, maintaining a
potential difference across the internal circuit. The value of
this potential difference is less than the open-circuit potential difference which, you will recall, was numerically
equal to the induced emf.
20.6 Lenz's Law
The downward motion of the copper
wire in Figure 20-4 is maintained by exerting a force on it
1-
II11-
ELECTROMAGNETIC
INDUCTION
~97
in the
action
closed
The
current
by the
direction of v. This force may be thought of as the
that generates the induced electron current in the
circuit.
relationship between the direction of an induced
and the action inducing it was recognized in 1834
German physicist Heinrich Lenz (1804-1865). His
discovery, now referred to as Lenz's law, is true of all induced currents. Because Lenz'slaw refers to induced currents, it applies only to closed circuits. Lenz's law states
that an induced current is in such a direction that it opposes the
change that induced it. Thus, the magnetic effect of the induced current in Figure 20-4 must be such that it opposes
the downward force (the pushing action) applied to the
conductor segment in the external field of the magnet.
Faraday discovered
how to deter-
mine the direction of an induced
current also, but he did not express it as clearly as Lenz.
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We can readily visualize what this effect must be. A
cross-sectional view of a conductor poised in a magnetic
field is shown in Figure 20-6(A). The conductor corresponds to end b of the wire in Figure 20-4. It is part of a
closed circuit that cannot be seen in this cross-sectional
diagram. As the wire is pushed downward through the
magnetic field, the induced electron current is directed
into the page as indicated by the x symbol (tail of the
arrow). By Ampere's rule for a straight conductor, the
magnetic field of this induced current encircles the current
in the counterclockwise sense. Similar diagrams in Figure
20-7 represent the effects of the wire being pulled up
through the magnetic field. Here the induced electron current is directed out of the page as indicated by the. symbol
(head of the arrow). The composite, or resultant, fields are
shown in Figures 20-6(B) and 20-7(B). They suggest that
the agent moving the conductor always experiences an
opposing force.
The fact that an induced current always opposes the
motion that induces it illustrates the conservation-ofenergy principle. Work must be done to induce a current
Figure 20-6. Two interacting magnetic fields. The separate magnetic lines of the fields of the
magnet and of the induced electron flow (into the page) are
shown in (A). The lines of the resultant field are shown In (8).
See Section 19.9 for Ampere's rule.
CHAPTER
498
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Figure 20-7. Two interacting magnetic fields. The separate magnetic lines of the fields of the
magnet and of the induced electron flow (out of the page) are
shown in (A). The lines of the resultant field are shown in (B).
QUESTIONS:
1"
in a closed circuit. The energy expended comes from outside the system and is a result of work done by the external force required to keep the conductor segment moving.
The induced current can produce heat or do mechanical or
chemical work in the external circuit as electrons of high
potential energy fall through a difference of potential.
GROUP A
1, What is an induced current?
2. What contribution did each of the following scientists make to our knowl.
edge about electromagnetic induction?
(a) Joseph Henry (b) Michael Faraday
(c) Hans Christian Oersted
3. What two conditions must be satisfied
to produce an electric current with a
magnetic field?
4. Referring to Ohm's law, upon what
factor besides induced emf does in~
duced current depend?
5. What three factors determine the
value of the induced emf?
6. State Lenz's law.
8.
9.
10.
GROUP R
7. Thrusting the north pole of a bar
magnet into a 600-tum coil of copper
wire causes a galvanometer attached
to the coil to deflect to the right.
What would happen if you: (a) Pull
11.
12.
13.
the magnet out of the coil? (b) Let the
magnet sit at rest in the coil? (c) Slide
the magnet left and right? (d) Turn
the magnet around and thrust the
south end of the magnet into the
coil?
A student is turning the handle of a
small generator attached to a lamp
socket containing a 15-W bulb. The
lamp barely glows. (a) What should
she do to make it glow more
brightly? (b) Why does this work?
What physical quantities are measured in (a) webers, (b) webers per
metei', (c) webers per second,
(d) joules per coulomb?
Explain how Lenz's law illustrates
conservation of energy.
Demonstrate that the product Blv is
properly expressed in volts.
Demonstrate that the product QvB is
properly expressed in newtons.
A permanent magnet is moved away
from a stationary coil as shown in the
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ELECTROMAGNE1K
INDUCTION
499
accompanying diagram. (a) What is
the direction of the induced current
in the coil? (Indicate direction of electron flow in the straight section of
conductor below the coil.) (b) What
magnetic polarity is produced across
the coil by the induced current?
(c) Justify your answers to (a) and (b).
PROBLEMS:
Permanent
magnet
75 cm/s. 1£the flux density is 0.025
GROUP A
1. A coil of 325 turns moving perpendicular to the flux in a uniform magnetic
field experiences a change in flux linkage of 1.15 X 10-5 weber
Motion
in 0.001 00 s.
What is the induced emf?
2. How many turns are required to produce an induced emf of 0.25 volt for a
coil that experiences a change in flux
Jinkage at the rate of 5.0 x 10-3 Wb/57
3. A straight conductor 10 em long is
moved through a magnetic field perpendicular to the flux at a velocity of
weber/m2, what emf is induced in the
conductor?
4. A coil of 75 turns and an area of
4.0 cm2 is removed from the gap between the poles of a magnet having a
uniform flux density of 1.5 \tVb/m2 in
0.025 s. What voltage is induced across
the coil?
5. A rod 15 em long is perpendicular
to a
magnetic
field of 4.5 x 10-1 N/A.
m
and is moved at rig!:it angles to the
flux at the rate of 30 cm/s. Find the
emf induced in the rod.
GENEHATORS AND MOTORS
20.7 The Generator Principle
An emf is induced in a
conductor whenever the conductor experiences a change
in flux linkage. When the conductor is part of a closed
circuit, an induced current can be detected in the circuit.
By Lenz's law, work must be done to induce a current in a
conducting circuit. Accordingly, this is a practical source
of electric energy.
Moving a conductor up and down in a magnetic field is
not a convenient method of inducing a current. A more
practical way is to shape the conductor into a loop, the
ends of which are connected to the external circui t by
means of slip rings, and rotate it in the magnetic field. See
Figure 20-8.
Such an arrangement is a basic generator. The loop
across which an emf is induced is called the armature. The
ends of the loop are connected to slip rings that rotate as
the armature is turned. A graphite brush rides on each slip
ring, connecting the armature to the external circuit. An
electric generator converts mechanicalenergy into electricenergy. The essential components of a generator are a field
magnet, an armature, and slip rings and brushes.
CHAPTER 20
500
Field magnet
,
\
.
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Figure 20-8. The essential components of an electric generator.
Figure 20-9.
atar rule.
The left-hand gener-
The induced emf across the armature and the induced
current in the closed circuit result from relative motion
between the armature and the magnetic flux that effects a
change in the flux linkage. Thus either the armature or the
magnetic field may be rotated. In some commercial generators the field magnet is rotated and the armature is the
stationary element.
According to Lenz's law, an induced current will appear
in such a direction that the magnetic force on the electrons
comprising the induced current opposes the motion producing it. The direction of induced current in the armature
loop of a generator can be easily determined by use of a
left-hand rule known as the generator rule. This rule takes
into account Lenz's Jaw and the fact that an e!edric current
in a metal conductor consists of a flow of electrons. See
Figure 20-9.
The generator rule: Extend the thumb, forefinger, and the
middle finger of the left hand at right ang!es to each other.
Let the forefinger point in the direction of the magnetic
flux and the thumb in the direction the conductor is moving; the middle finger points in the direction of the induced electron flow.
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II1-
ELECTROMAGNETIC
INDUCTION
501
20.8 The Basic a-c Generator
The two sides of the conducting loop in Figure 20-8 move through the magnetic
flux in opposite directions when the armature is rotated.
By applying the generator rule to each side of the loop, the
direction of the electron flow is shown to be toward one
slip ring and away from the other. Thus a single-direction
loop is established in the closed circuit. As the direction of
each side of the loop changes with respect to the flux, the
direction of the electron flow is reversed. As the armature
rotates through a complete cycle, there are two such reversals in direction of the electron flow.
In Figure 20~10 one side of a conducting loop rotating in
a magnetic field is shown cross-sectionally in color while
the other side is shown in black. In (A) the colored side of
the loop is shown moving down and cutting through the
flux. By the generating rule, the electron flow is directed
into the page. (The white x represents the tail of the
arrow.) The motion of the black side of the loop induces a
fJow directed out of the page. (The white dot represents
the head of the arrow.)
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An emf is induced in a conductor as a result of a change
in the flux linking the conductor. In (B) both sides of the
loop are moving parallel to the flux and there is no change
in linkage. Therefore, no emf is induced across the loop
and there is no electron flow in the dosed circuit.
In (C) the black side of the loop is moving down and
cutting through the flux so that the electron flow is directed into the page. The colored side is moving up
through the flux, and thus the electron flow is directed out
of the page. The direction of the flow in the circuit is the
reverse of that in (A). A quarter cycle later, the emf again
drops to zero and there is no flow in the circuit. The emf
induced across the conducting loop reaches a maximum
value when the sides of the loop are moving perpendicular to the magnetic flux. see Section 20.2.
We can see from Figure 20-10 that the magnitude of the
induced emf across the conducting loop must vary from
zero through a maximum and back to zero during a half
Figure 20-10. An emf is induced
only when there is a change in the
flux linking a conductor.
CHAPTER
;)02
20
cycle of rotation. The emf must then vary in magnitude in
Figure 20-11. One cycle of operation of an a-c generator.
a similar manner, from zero through the same magnitude
maximum and back to zero, during the second half cycle.
However, polarity across the loop is opposite. The emf
across the loop thus altemates in polarity.
Similarly, the flow in a circuit connected to the rotating
armature by way of the slip rings alternates in direction:
electrons flowing in one direction during one half cycle
and in the opposite direction during the other half cycle. A
currellt that lias one direction durillg part of a generatingcycle
and the opposite direction durins the remail/der of tile cycle is
called an alternating current, I1C.A generator that produces
an alternating current must, of course, produce an alternating emf. See Figure 20-11.
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Commercial electric power is usual1y supplied by the
generation of alternating currents and voltages. Such
power is referred to as n-c power. The expressions at and
de are often used to distinguish
between alternatingcurrent and direct-current properties: a-( voltage and d~c
current are two examples.
M---
Figure 20-12. The instantaneous
value of an induced voltage varies
with the sine of the displacement
angle of the loop in the magnetic
field.
20.9 Instantaneous
Current and Voltage
The opencircuit voltage across a battery has a constant magnitude
characteristic of the chemical makeup of the battery. The
voltage across an armature rotating in a magnetic field,
however, has no constant magnitude. It varies from zero
through a maximum in one direction and back to zero during one half cycle. It then rises to a maximum in the opposite direction and falls back to zero during the other half
cycle of armature rotation. At successive instants of time,
different magnitudes of induced voltage exist across the
rotating armature. The magnitude of a VIlnJing voltage at allY
instant of time is called the instantaneous voltage, e.
The maximum voltage, ~m"'" is obtained v.,rhen the conductor is moving perpendicular to the magnetic flux because the rate of change of flux linking the conductor is
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ELECTROMAGNETIC INDUCTION
50.1
maximum during this time. If the armature is rotating nt a
constant rate in a magnetic field of uniform flux dCl1sity, the
magnitude of the induced voltage varies sinusoidally (as a sine
wave) with respect to time.
In Figure 20-12 a single loop is rotating in a uniform
magnetic field. When the plane of the loop is perpendicu~
lar to the flux (MN of Figure 20-12), the conductors are
moving parallel to the flux lines; the displacement angle of
the loop is said to be zero. We shall refer to this angle
between the plane of the loop and the perpendicular to the
magnetic flux as 0 (theta). When fJ = 0° and 180°, e = 0 V.
When (j = 90°, e = 't;m~x; and when () = 270°, e =_C&max'
These relationships
are apparent from Figure 20-11. In
general, the instantaneous
voltage, e, varies with the sine
of the displacement angle of the loop.
e = ~rna" sin (J
The current in the external circuit of a simple generator
consisting of pure resistance will vary in a similar way; the
maximum current, In"I}(;occurs when the induced voltage is
maximum. From Ohm's law
'jgma"
Irna"=~
R
The instal1taneolls current, i, is accordingly
e
i=R
but
e = ~ma" sin (J
so
i = '" m.." sin (J
R
and
20.10 Practical a-c Generators
The simple generator
consists of a coil rotating in the magnetic field of a permanent magnet. Any small generator employing a permanent magnet is commonly called a magneto. Magnetos are
often used in the ignition systems of gasoline engines for
lawn mowers, motorbikes, and boats.
The generator output is increased in a practical generator by increasing the number of turns on the armature or
increasing the field strength. The field magnets of large
generators arc strong electromagnets;
in a-c generators
they are supplied with direct current from an auxiliary d-c
generator called an exciter. See Figure 20-13.
The performance of large a-c generators is generally
more satisfactory jf the armature is stationary and the field
rotates inside the armataTe~'-Such stationary armatures are
Conductof5fo.
a-coutput
[d-Cgenerator
fo.magnetizing
the field
Figure 20-13. The field of a large
alternating-current
generator is
produced by an electromagnet.
CHAPTER 20
504
+
t
Position
Li
Figure 20-14.
senting current
A sine curve repre-
or voltage gener-
ated by a single-loop armature
rotating at a constant rate in a
uniform magnetic field. Positions
o through 12 on the graph correspond to positions of the rotating
armature in the magnetic field. as
shown to the left.
referred to as stators and the rotating field magnets as rotors. Circuit current is taken from the stator at the high
generated
voltage without the use of slip rings and
brushes. The exciter voltage, which is much lower than
the armature voltage, is applied to the rotor through slip
rings and brushes.
In a simple two-pole generator one cycle of operation
produces one cycle or two alternations of induced emf, as
shown in Figure 20-14. If the armature (or the field) rotates
at the rate of 60 cycles per second, the frequency, f, of the
generated voltage sine wave is 60 hertz and the period, T,
is
The "frequency" of a d-c voltage
or current is considered to be
zero.
Figure 20-15. The principle of the
three-phase alternator. Armature
schematically diagrammed in
(A) are spaced 1200 apart and
generate peak voltage output in
three phases as shown in (B). In
(C) the output is shown as a voltage diagram.
coils
0
O.
610
second.
The frequency of an alternating current or voltage is the num~
ber of cycles of current or voltage per second. If the generator
has a 4-pole field magnet, 2 cycles of emf are generated
during 1 revolution of the armature or field. Such a generator turning at 30 rps would generate a 60-Hz voltage. In
general
f = No. of pairs of poles x revolution rate
Practically all commercial power is generated by threephase generators having three armature coils spaced symmetrically and producing emfs spaced 1200 apart. The coils
are usually connected so that the currents are carried by
three conductors.
,
c
B
t
o
,
-" O.
,
1200
,,
240.
(B) Voltage output
,
,
360.
A
(C) Voltage diagram
.
r
ELECTROMAGNETIC INDUCTION
505
It is evident from Figure 20-15(B) that three-phase
power is smoother than the single-phase power of Figure
20-14. Electric power is transmitted by a three-phase circuit but it is commonly supplied to the consumer by a
single~phase circuit. A modern center for the control of
production and distribution of electric energy is shown in
Figure 20-16.
20.11 The d-c Generator The output of an a-c generator is not suitable for circuits that require a direct current.
However, the a-c generator can be made to supply a unidirectional current to the external circuit by connecting the
armature loops to a commutator instead of slip rings. A
commutator is a split ring, each segment of which is connected
to an end of a corresponding
armature
loop.
The current and voltage generated in the armature are
alternating, as we would expect. By means of the commutator, the connections to the external circuit are reversed at
the same instant that the direction of the induced emf reverses in the loop. See Figure 20-17. The alternating current in the armature appears as a pulsating direct current in
the external circuit, and a pulsating d-c voltage appears
across the load. A graph of the instantaneous values of the
pulsating current from a generator with a two-segment
commutator plotted as a function of time is shown in Figure 20-18. A graph of the voltage across a resistance load
would have a similar form. Compare this pulsating d-c
output to the a-c output of the generator in Figure 20-14.
To secure from a d-c generator a more constant voltage
and one having an instantaneous
value that approaches
the average value of the emf induced in the entire armature, many coils are wound on the armature. Each coil is
connected to a different pair of commutator segments. The
two brushes of each commutator segment are positioned
so that they are in contact with successive pairs of commutator segments at the time when the induced emf in their
respective coils is in the 'fmax region. See Figure 20-19.
20.12 Field Excitation
Most d-c generators use part of
the induced power to energize their field magnets and are
said to be self-excitirlg. The field magnets may be connected
in series with the armature loops so all of the generator
current passes through the coil windings. In the serieswound generator an increase in the load increases the magnetic field and hence the induced emf. See Figure 20-20.
The field magnets may also be connected in parallel
with the armature so only a portion of the generated current is used to excite the field. In this shunt-wound generator, Figure 20-21, an increase in load results in a decrease
in the field and hence a decrease in the induced emf.
Figure 20-16. The control room of
an electric power generating station. Computers monitor power
production and transmission.
B
Figure 20-17. A split-ring commutator of two segments.
e.i
o
"'0
Figure 20-18. The variation of current or voltage with time in the
external circuit of a simple
generator with a two-segment
commutator.
CHAPTER 20
.';06
,
Coil Coil Coil
1112,31
1 I 2 13
.,
"r.. / "I.. /1<- / /hI , / /1'- / "I1
'../1'
j( 1 X 1I' )r I' X I )f I' X 1
1 ,,1
1/,\ " I
/ " '\,I,1/' \1/
'1/ '" ,', " ,I,
\It
o
50' 120' 180' 240' 300. ~'
Onecompletecy<;le~
Figure 20-19. The output of a d-c
generator having three armature
coils and a six-segment commutator is fairly constant.
By using a combination of both series and shunt windings to excite the field magnets, the potential difference
across the external circuit of a d-c generator may be maintained fairly constant; an increase in load causes an increase in current in the series windings and a decrease in
current in the parallel windings. With the proper number
of turns of each type of winding, a constant flux density
can be maintained under varying loads. A compoundwound generator is shown in Figure 20-22.
20.13 Ohm's Law and Generator
Circuits
We can
think of the armature turns as the source of emf in the d-c
generator circuit. If the field magnet were separately excited, this emf would appear across the armature terminals on open-circuit operation since there would be no
induced armature current.
,-
111:,,,,,,,,,,,
Load
-,Figure 20-20.
generator.
load
!
Armature
+
(A) Pictorial diagram
,+
Sliunt
field
load
,Figure 20-21. A shunt-wound d-c
generator.
+
load
(A) Pictorial diagram
However, in a self-excited generator the armature circuit is completed through the field windings. The resistance of the armature turns, rAfis in this current loop, and a
situation analogous to that of a battery with internal resistance furnishing current to an external circuit results. A
potential drop, lAr", of opposite polarity to the induced
/-"
r'
ELECTROMAGNETIC INDUCTION
507
-
,+
SllUnt
field
Load
+
,-~
+
Load
(A) Pictorial diagram
emf, must appear across the armature.
tential difference, V, is
V= ~
r
-
The armature
po-
which has been shown to be W,- lAr ATthen appears
across
the network consisting of Rf and RLin parallel. The following Ohm's law relationships hold:
h=-
and
A compound-wound
lArA
Figure 20-23 is a resistance circuit of a series~wound generator. The resistance of the field windings, RF1is in series
with the internal resistance of the armature and the load
resistance so that the armature current is present in all
three resistances.
In Figure 20-21, the resistance of the shunt windings is
in parallel with the load. The resistance circuit of a shuntwound generator is diagrammed in Figure 20-24, with the
resistance of the shunt windings shown as RIo The total
current of the circuit, 1/11must be in r AI producing
an lArA
drop across the armature. The potential difference, V,
IF=-
Figure 20-22.
doc generator.
A
B
Figure 20-23. The resistance
circuit of a series-wound
doc
generator.
V
R .'
v
A
Rc
L = IF + lL
The total electric power in the generator circuit, PH derived from the mechanical energy source that turns the
armature, is the product of the armature current, 1M and
the induced emf, '~.
Some of this power is dissipated
as heat in the armature
R"
B
Figure 20-24. The resistance
circuit of a shunt-wound d-c
generator.
CHAPTER 20
508
resistance, and some is dissipated in the field windings;
the remaining power is delivered to the load.
P, = p" + Pr + P,.
A
R"1
'f,J", = IA'2rA+ I/RF + IL2R1
0'
.
B
Figure
20-25.
The resistance
cuit of a compound-wound
generator.
d-c
cir-
The application of these relationships to the d-c shuntwound generator is illustrated in the example that follows.
Figure 20-25 is a resistance circuit of a compoundwound generator. Observe the similarity between this circuit and the resistance network shown in Figure 17-21,
Section 17.10. How would you analyze this resistance circuit of the compound-wound d-c generator?
EXAI\'lPLE
A shunt-wound
d-c generator has an armature resistance
of 0.45 fl. The shunt has a resistance of 65.2 n. The generator delivers
1250 W to a load with a resistance of 39.0 U. Calculate (a) the voltage
across the load, (b) the current in the armature, and (c) the emf of the
generator.
I.
I
Solution
V,'
(a) Working equation: PL = -
V,
R
~
V(P,)(R,)
~
221 V
~
V(1250 W)(39.0 f!)
(b) The armature current, 1M is the sum of the shunt current, IF, and the
current in the load, IL" Using Ohm's law, we get the
War
.
VF
VI.
k llJer
'
equatIOn: I = - + 'R
R'.
'
.
"
221 V
221 V
+
39.0 f!
65.2 f!
(c) Working equation: '8
=V+
~
~
9.06 A
lArA
221 V + (9.06 A)(0.45 f!) ~ 2.3 x to' V
ELECTROMAGNETIC INDUCTION
,
,';09
PRACTICI~ l-nOBLEM
A shunt-wound
d-c generator has an armature
resistance of 0.35 n and a field-winding resistance of 75.0 n. The generator delivers 1.44 kW to a load resistance of 22.5 n. (a) Draw the circuit
diagram. (b) What is the potential difference across the load? (c) What is
the magnitude of the armature current? (d) Wha.! is the emf of the generator?
Ans. (a) See Figure 20-24; (b) 180 V; (c) 10.4 A; (d) 184 V
20.14 The Motor Effect
We have learned that a current
is induced in a conducting loop when the conductor is
moving in a magnetic field so that the magnetic flux linking the loop is changing. This is the generator principle.
By Lenz's law, we have seen that work must be done
against a magnetic force as an induced current is generated in the conducting loop. Recall that an induced current
always produces a magnetic force that opposes the force
causing the motion by which the current is induced.
,
,
Figure 20-26. A visual representation of the motor effect.
Expul.ionforce
f
f
r'.
r
IAI
I"
Instead of employing a mechanical
effort to move a conductor poised in the magnetic field, suppose a current is
supplied to the conductor from an external source. The
magnetic field is distorted, as discussed in Section 20.6,
and the resulting magnetic force tends to expel the conductor from the magnetic field. This action is known as the
motor effect, and it is illustrated in Figure 20-26.
If a current is supplied to an armature loop poised in a
uniform magnetic field, as in Figure 20-27, the field
around each conductor is distorted. A force acts on each
side of the loop proportional to the flux density and the
current in the armature loop. These two forces, equal in
magnitude but opposite in direction, constitute a couple
and produce a torque, T, that causes the armature loop to
rotate about its axis. The magnitude of this torque is equal
to the product of the force and the perpendicular distance
between the two forces.
F
-"
F
Figure 20-27. The resultant magnetic field of a current loop in the
external field of a magnet gives a
visual suggestion of the forces
acting on the loop.
CHAPTER
510
It is evident from Figure 20-27 that the perpendicular
distance between the torque-producing
forces is maximum when the conductors are moving perpendicular
to
the magnetic flux. This distaI"Lce is equal to the ..\'idth of
the armature loop. When the loop is in any other position
with respect to the flux lines, the perpendicular distance
between the forces is less than the \,.,ridth of the loop, and
consequently the resulting torque must be less than the
maximum value. See Figure 20-28.
We shall refer to the angle between the plane of the loop
and the magnetic flux as angle a (alpha). When the angle
is zero, the plane of the loop is parallel to the flux lines and
the torque is maximum.
F
I
+I
N
Tmax
Figure 20-28. The torque on a
current
loop
20
= Fw
where F is the magnetic force acting on either conductor
and IV is the width of the conducting loop.
As the armature turns, the angle a approaches 90" and
the torque diminishes since the perpendicular
distance
between the couple approaches zero. In general, the perpendicular distance between the forces acting on the two
conductors is equal to IV cas a, as shown in Figure 20-28.
Hence
in a magnetic field is
proportional
to the perpendicular
distance between the forces acting on the conductors.
T=FwcoslX
When the plane of the loop is perpendicular to the magnetic flux, angle a is 90", the cosine of 90" = 0, and the
torque is zero. As the inertia of the conductor carries it
beyond this point, a torque develops that reverses the
motion of the conductor and returns it to the zero-torque
position. In order to prevent this action, the direction of
the current in the armahtre loop must be reversed at the
proper instant. To reverse the current when the neutral
position is reached, the conducting loop terminates in a
commutator.
An electric motor performs the reverse function of a
Motion
t
generator.
I
Electric
energy
is converted
to II1ccJumiCClI CIlergy
using the same electromagnetic
principles employed in
the generator. We can determine the direction of the motion of the conductor on a motor armature by use of a
right-hand rule known as the motor rule. It is jlJustrated in
Figure 20~29.
The motor rule: Extend the thumb, forefinger, and middle
finger of the right hand at right angles to each other. Let
the forefinger point in the direction of the magnetic flux
and the middle finger in the direction of the electron flow;
Figure 20-29. The right-hand
the thumb points in the direction of the motion of the con-
motor rule.
ductor.
"
r
,
ELECTROMAGNETIC
511
INDUCTION
c
c
,
,
,
,
f
I
f
I
r
I
I
I
I
I
(A) Motor stalled
20.15 Back emf The simple d-c motor does not differ
essentially from a generator; it has a field magnet, an ar-
mature, and a commutator ring. In fact, the operating motor
also acts as a generator. As the conducting loop of the armature rotates in a magnetic field, an emf is induced across
the armature turn. The magnitude of this emf depends on
the speed of rotation of the armahtre.
Suppose that an incandescent lamp and an ammeter are
connected in series with the armature of a small batterydriven motor, as illustrated in Figure 20-30. If the motor
armature is held so that it cannot rotate as the circuit is
closed, the lamp glows and the circuit current is indicated
on the meter. Releasing the armature allows the motor to
gain speed, and the lamp dims; the ammeter indicates a
smaller current.
According to Lenz's law, the induced emf must oppose
the motion inducing it. The emf induced by the generator
action of a motor consequently opposes the voltage applied to the armature. Such an induced emf is calledthe back
emf of the motor. The difference between the applied voltage and the back emf determines the current in the motor
circuit.
A motor running at full speed under no load generates a
back emf nearly equal to the applied voltage; thus a small
current is required in the circuit. The more slowly the armature turns, the smaller is the back emf and consequently the larger is the voltage difference and the circuit
(8) Motor running
Figure 20-30. Demonstrating the
back emf of a motor.
CHAPTER 20
512
current. A motor starting under a full load has a large initial current that decreases due to the generation of a back
emf as the motor gains speed.
The induced emf in a generator is equal to the terminal
voltage plus the voltage drop across the armature resistance.
In the motor, the induced emf is equal to the terminal voltage minus the voltage drop across the armature resistance.
~ = V - LrA
Hence, the back emf in a motor must always be less than
the voltage impressed across the armature terminals.
20.16 Practical d-c Motors
A simple motor with a single armature coil would be impractical for many purposes
because it has neutral positions and a pulsating torque. In
practical motors a large number of coils is used in the armature; in fact there is little difference in the construction
of motor and generator armatures.
Multiple-pole
field
coils can be used to aid in the production of a uniform
torque. The amount of torque produced in any given
motor is proportional to the armature current and to the
flux density.
Depending on the method used to excite the field magnets, practical d-c motors are of three general types. These
are series, shunt, and compound motors. The excitation
methods are similar to those of the d-c generators discussed in Section 20.12.
20.17 Practical a-c Motors
Nearly all commercial distributors of electric power supply alternating-current
power. Except for certain specialized applications,
a-c
motors are far more common than d-c motors. Much of the
theory and technology of a-c motors is complex. We will
briefly summarize the general characteristics of three common types of a-c motors: the universal motor, the induction
motor, and the synchronous motor.
1. The universal motor. Any small d-c series motor can be
operated from an a-c source. When this is done, the currents in the field windings and armature reverse direction
simultaneously,
maintaining torque in the same direction
throughout the operating cycle. Heat losses in the field
windings are extensive, however, unless certain design
changes are incorporated into the motor. With laminated
pole pieces and special field windings, small series motors
11-
ELECTROMAGNETIC
INDUCTION
operate satisfactorily on either a-c or d-c power. They are
known as universal motors.
2. The induction motor. Induction motors are the most
widely used a-c motors because they are rugged, simple to
build, and well adapted to constant speed requirements.
They have two essential parts-a
stator of field coils and a
rotor. The rotor is usually built of copper bars laid in slotted, laminated iron cores. The ends of the copper bars are
shorted by a copper ring to form a cylindrical cage; this
common type is known as a squirrel-cage rotor.
By using three pairs of poles and a three-phase current,
the magnetic field of the stator is made to rotate electrically
and currents are induced in the rotor. In accordance with
Lenz's law, the rotor will then turn so as to follow the
rotating field. Since induction requires relative motion between the conductor and the field, the rotor must slip or
lag behind the field in order for a torque to be developed.
An increase in the load causes a greater slip, a greater induced current, and consequently a greater torque.
For its operation, an induction motor depends on a rotating magnetic field. Thus a single-phase induction motor
is not self-starting because the magnetic field of its stator
merely reverses periodically and does not rotate electrically. Single-phase induction motors can be made selfstarting by a split-phase winding, a capacitor, a shading coil, or
a repulsion winding. The name of the motor usually indicates the auxiliary method used for starting it.
3. The synchronous motor. We can illustrate the principle
of synchronism by placing a thoroughly magnetized compass needle in a rotating magnetic field. The magnetized
needle aligns itself with the magnetic field and rotates in
synchronization
with the rotating field.
The synchronous motor is a constant speed motor, running in synchronism with the a-c generator that supplies
the stator current. However, it is not self-starting; the
rotor must be brought up to synchronous speed by auxiliary means.
Electric clocks are operated by small single-phase synchronous motors that start automatically as a form of induction motor. Once synchronism is attained, they ron as
synchronous motors. Electric power companies maintain
very accurate control of the 60-Hz frequency of commercial power.
Large industrial synchronous motors use an electromagnetic rotor supplied with d-c power, and a stator supplied
with three-phase a-c power. The synchronous motors are
used where the speed requirements are very exacting.
513
Figure 20-31. A cut-away view of
an induction motor.
CHAPTER
.';14
QUES110NS:
GROUP A
1. What are the essential components of
an electric generator?
2. State the rule that helps us determine
the direction of the induced current
in the armature loops of a generator.
3. Distinguish between a direct current
and an alternating current.
4. Under what circumstances does a
simple generator produce a sine-wave
variation of induced voltage?
5. What does the term (J(theta) repre-
sent in the expression e = 'jgmax sin O?
6. (a)What is a magneto?
it used?
(b) For what is
7. What is the function of an exciter in
the generation of a-c power?
8. (a) What is meant by the frequency of
an alternating current? (b) Under
what circumstances will the frequency
of a generated current be the same as
the rps of the armature?
9. How can a generator be made to supply a direct current to its external circuit?
10. In what way is the output of a d-c
generator different from the d-c out.
put of a battery?
11. (a) What methods arc used to energize the field magnets of d-c generators? (b) Draw a circuit diagram of
each method.
12. What are the three power-consuming
parts of a d-c generator circuit?
13. What is meant by the term a (alpha)
in the torque expression T =
Fw cas a?
14. State the rule that helps us determine
the direction of motion of the armature loops of a motor.
15. (a) What is meant by back emf?
(b) How is it induced in an electric
motor?
16. What two quantities influence the
amount of torque produced in a
motor?
17. What are the three common types of
a-c motors?
20
GROUP B
18. What is the advantage of having a 4or a 6-pole field magnet in a generator producing a 60-Hz output?
19. How does an increase in the load on
a series-wound d-c generator affect
the induced emf? Explain.
20. How does an increase in the load on
a shunt-wound
d-c generator affect
the induced emf? Explain.
21. How is torque produced on the armature loops of a motor?
22. Why is it true that an operating
motor is also a generator?
23. The torque in a series motor increases
as the load increases. Explain.
24. Explain why a single-phase induction
motor is not self-starting.
25. Why are synchronous motors used in
electric clocks?
26. Two conducting Joops, identical except that one is silver and the other
aluminum, are rotated in a magnetic
field. In which case is the larger
torque required to turn the loop?
PROBLEMS:
GROUP A
1. A series-wound d-c generator turning
at its rated speed develops an emf of
28 V. The current in the external circuit
is 16 A and the armature resistance is
0.25 H. What is the potential drop
across the external circuit?
2. A d-c generator delivers 150 A at 220 V
when operating at normal speed con.
nected to a resistance load. The total
losses are 3200 W. Determine the efficiency of the generator.
3. A shunt.wound
generator has an armature resistance of 0.15 n and a
shunt winding of 75.0-fl resistance. It
delivers 18.0 kW at 240 V to a load.
(a) Draw the circuit diagram. (b) What
is the generator emf? (c) What is the
total power delivered by the armature?
4. A magnetic force of 3.5 N acts on one
conductor of a conducting loop in a
~
,
I
ELECTROMAGNETIC
INDUCTION
magnetic field. A second force of equal
magnitude but opposite direction acts
on the opposite conductor of the loop.
The conducting loop is 15 em wide.
Find the torque acting on the conducting Joop (a) when the plane of the loop
is parallel to the magnetic flux, (b)
after the loop has rotated through 30°.
5. The maximum torque that acts on the
armature loop of a motor is 10 ill . N.
515
At what positions of the loop with respect to the magnetic flux will the
torque be 5 ill . N?
6. A shunt-wound
motor connected
across a 117-V line generates a back
emf of 112 V when the armature current is 10 A. What is the armature resistance?
INDUCTANCE
20.18 Mutual Inductance
An emf is induced across a
conductor in a magnetic field when there is a change in the
flux linking the conductor. In our study of the generator
we observed that an induced emf appears across the armature loop whether the conductors move across a stationary
field or the magnetic flux moves across stationary conductors. In either action there is relative motion betweenconductors
and magnetic flux.
This relative motion can be produced in another way.
By connecting a battery to a solenoid through a contact
key, an electromagnet is produced that has a magnetic
field similar to that of a bar magnet. When the key is open,
there is no magnetic field. As the key is closed, the magnetic field builds up from zero to some steady value determined by the number of ampere-turns.
The magnetic flux
spreads out and permeates the region about the coil. An
expanding magnetic field is a field in motion. When the key in
the solenoid circuit is opened, the magnetic flux collapses
to zero. A collapsing magnetic field is also a field in motion. Its
motion, however, is in the opposite sense to that of the
expanding field.
Suppose the solenoid is inserted into a second coil
whose terminals are connected to a galvanometer as in
Figure 20-32. The coil connected to a current source is the
primary coil; its circuit is the primary circuit. The coil connected to the galvanometer (the load) is the secondarycoil;
its circuit is the secondary circuit. At the instant the contact
key is closed, a deflection is observed on the galvanometer. There is no deflection, however, while the key remains closed. When the key is opened a galvanometer
Figure 20-32. Varying the current
deflection again occurs, but in the opposite direction. An in the primary induces an emf in
emf is induced across the secondary turns whenever the the secondary.
516
CHAPTER
20
flux linking the secondary is increasing or decreasing.
The relative motion between conductors and flux is in
one direction when the field expands and in the opposite
direction when the field collapses. Thus the emf induced
across the secondary as the key is closed is of opposite
polarity to that induced as the key is opened. The more
rapid this relative motion, the greater the magnitude of
the induced emf; the greater the number of turns in the
secondary, the greater the magnitude of the induced emf.
A soft-iron core placed in the primary greatly increases the
flux density and the induced emf.
Two circuits arranged so that a change in magnitude of
current in one causes an emf to be induced in the other
show mutual inductance. The mutual inductance, M, of two
circuits is the ratio of the induced emf in one circuit to the ratc of
change of current in the other circuit.
The unit of mutual inductance is called the hennj, H,
after the American physicist Joseph Henry. The /Ill/trial inductance of two circuits is onc henry wilen OIlCuo/t of Clllf is
iwiuced in the sccondary as the current in the primary clulIIgesat
the rate of one ampere per second.
M~
-~,
IJ.I,./lJ.t
Here, M is the mutual inductance of the two circuits in
henrys, '¤osis the average induced emf across the secondary in volts, and l1Ip/l1t is the time rate of change of current
in the primary in amperes per second. The negative sign
indicates that the induced voltage opposes the change in
current according to Lenz's law. From this equation the
emf induced in the secondary becomes
'I:, ~-M-
Figure 20-33. Joseph Henry, the
American physicist for whom the
unit of inductance is named.
illp
IJ.t
See the following example.
EXA\WI.E
When the switch is dosed in the circuit of the primary coil
of a pair of induction coils, the current reaches 9.50 i! in 0.032 0 s. The
average voltage induced in the secondary coil is 450 V. Calculate the
mutual inductance of the two coils.
Unknown
.>1,,~9,50 A
.>, ~ 0,032 0 s
'I:,~450 V
M
,
.'i17
ELECTROMAGNETIC INDUCTION
Solution
Working
equation:
M
=
,
c
"
I,
e
f
r
[
-%s !it
Ie
-(450 V)(0.0320 s)
9.50 A
~
-1.52 H
PIU(:TlCE IJR()IUEMS 1. Two adjacent coils have a mutual inductance
of 0.880 H. What is the average induced emf in the secondary if the
current in the primary changes from 0.00 A to 18.0 A in 0.055 0 s?
Ans. -288 V
2. The current in the primary of two adjacent coils changes from 0.0 A to
14 A in 0.025 s and the average induced emf in the secondary is 320 V.
What is the mutual inductance of the two circuits?
Ans. 0.57 H
20.19 Self-Inductance
Suppose a coil is formed by
winding many turns of insulated copper wire on an iron
core and connected in a circuit to a 6-volt battery, a neon
lamp, and a switch. See Figure 20-34. Since the neon lamp
requires about 85 V de to conduct, it acts initially as an
open switch in a 6-volt circuit.
When the switch is closed, a conducting path is completed through the coil; the fact that the lamp does not
light is evidence that it is not in the conducting circuit. If
now the switch is quickly opened, the neon lamp conducts
for an instant, producing a flash of light. This means that
the lamp is subjected to a potential difference considerably
higher than that of the battery. What is the source of this
higher voltage?
Any change in the magnitude of current in a conductor
causes a change in the magnetic flux about the conductor.
If the conductor is formed into a coil, a changing magnetic
flux about one turn cuts across adjacent turns and induces
a voltage across them. According to Lenz's law, the polarity of this induced voltage acts to oppose the motion of the
flux inducing it. The sum of the induced voltages of all the
turns constitutes a counter emf across the coil.
If a rise in current with its expanding magnetic flux is
responsible for the counter emf, this rise in current will be
sw
N~"
lamp
Figure 20-34. A circuit to demonstrate self-inductance.
518
It is not the magnitude
CHAPTER 20
of current
at which current is
changing that relates to the magnitude of an induced emf.
but the rate
opposed. Therefore, the counter emf is opposite in polarity to the applied voltage. When the switch in Figure 20-34
was closed, the rise in current from zero to the steadystate magnitude was opposed by the counter emf induced
across the coil. Once the current reached a steady value,
the magnetic field ceased to expand and the opposing
voltage fell to zero.
If a fall in current with its collapsing flux is responsible
for the induced voltage across the coil, the fall in current is
opposed. In this case the induced emf has the same polarity as the applied voltage and tends to sustain the current
in the circuit. A very rapid collapse of the magnetic field
may induce a very high voltage. It is the change in current,
not the current itself, that is opposed by the induced emf. It follows, then, that the greater the rate of change of current in
a circuit containing a coil, the greater is the magnitude of
the induced emf across the coil that opposes this change of
current.
The property of a coil that causes a counter emf to be
induced across it by the change in current in it is known as
self-inductance, or simply inductance. The self-inductance, L,
of a coil is the ratio of the induced emf across the coil to the rate of
change of current in the coil.
The unit of self-inductance is the henry, the same unit
used for mutual inductance. Self-inductance is one henry if
one volt of emf is induced across the coil when the current in
the circuit changes at the rate of one ampere per second.
-~
ill/tJ.t
L~-
Here L is the inductance in henrys, ~ is the emf induced in
volts, and &IIM is the rate of change of current in amperes
per second. The negative sign merely shows that the induced voltage opposes the change of current. From this
expression, the induced emf is expressed by
~
~
-L-
ill
M
The equation for inductance,
fining equation for capacitance,
16.14.
L, is analogous
C, as expressed
to the dein Section
c~R
V
The presence of a magnetic field in a coil conducting an
electric current corresponds to the presence of an electric
field in a charged capacitor.
..~
,
,
f
ELECTROMAGNETIC
.519
INDUCTION
Once a coil has been wound, its inductance
is a constant
property that depends on the number of turns, the diameter of
the coil, the length of coil, and the nature of the core. Because of
its property of inductance, a coil is commonly caJled an
inductor.
Inductance in electricity is analogous to inertia in mechanics: the property of matter that opposes a change of
velocity. If a mass is at rest, its inertia opposes a change
that imparts a velocity; if the mass has a velocity, its inertia
opposes a change that brings it to rest. The flywheel in
mechanics illustrates the property of inertia. We know that
energy is stored in a flywheel as its angular velocity is increased and is removed as its angular velocity is decreased.
In an electric circuit inductance has no effect as long as
the current is steady. An inductance does, however, oppose any change in the circuit current. An increase in current is opposed by the inductancei energy is stored in its
magnetic field since work is done by the source against the
counter emf induced. A decrease in current is opposed by
the inductancei energy is removed from its field, tending
to sustain the current. We can think of inductailCe as imparting
a flywheel effect in a circuit having a varying current. This concept is very important in alternating-current
circuit considerations.
20.20 Inductors
in Series and Parallel
The total inductance of a circuit consisting of inductors in series or
parallel can be calculated in the same manner as total resistance. When inductors are connected in series, the total
inductance, LT' is equal to the sum of the individual inductances providing there is no mutual inductance between them.
LT
=
Lt + Lz + L3 + etc.
However, when two inductors are in series and arranged
so that the magnetic flux of each links the turns of the
other, the total inductance is
The
j:
sign is necessary in the general expression for the
counter emf induced in one coil by the flux of the other may
either aid or oppose the counter emf of self-induction.
The two coils can be connected either in series "aiding"
or series "opposing," depending on the manner in which
their turns are wound.
In terms of mechanica/ qualltities,
impedann' is analogolls to mass,
and currmt is analogous to
ve/ocit!j.
520
CHAPTER 20
Inductors
connected
in parallel
so that
each
is unaf-
fected by the magnetic field of another provide a total inductance according to the folJowing general expression:
1
1
-=-+-+-+etc.
L,
L,
L,
LT
L,
See
connections
20.21 The Transfonner
L,
L3-
of these se-
for inductors.
In principle the transformer
electrically
insulated
magnetic
(')
Figure 20-35. Inductors
(A) and in parallel (B).
Lz
consists of two coils, a primary and a secondary,
,
o
1
Figure 20-35 for schematic representation
ries and parallel
(A)
L1
1
in series
from each other and wound on the same ferrocore. Electric energy is transferred
from the primary to the secondary by means of the magnetic flux in
the core. A transformer is shown in its simplest forms in
Figure 20-36_
load
load
Low-v oltage
'oco ndary
High-voltage
secondary
I
I
,
Low-voltage
primary
Figure 20-36. The transformer in
its simplest forms.
a-cgenerator
High -voltage
pri mary
'"
a-cgenerator
Refer to Figure 20~14, in which the current sine wave
produced by the a-c generator is given. The current is
changing at its maximum rate as it passes through zero.
Consequently,
the emf induced across the secondary
winding of a transformer is maximum as the primary current passes through zero. The polarity of the secondary
emf reverses each time the primary current passes
through a positive or negative maximum, since at these
instants the current is changing at its minimum rate.
In a power transformer, a closed core is used to provide a
continuous path for the magnetic flux, ensuring that practically all the primary flux links secondary turns. Since the
same flux links both primary and secondary turns, the
same emf per turn is induced in each and the ratio of secondary to primary emf is equal to the ratio of secondary to
.
ELECTROMAGNETIC
primary turns. Neglecting losses, we may assume the terminal voltages to be equal to the corresponding
emis.
Thus
Vs
=
Vp
n
::!i
. .::__c",--""",
,~. -.i
>""j.'.;lJl:,t
?\i
.
~""ir"~
~
..Li
~
I
- 1
~tl
r
ratio of 20.
Turns ratio = -
Ns
20
= - = 20
Np
1
If the primary voltage is 110 volts, the secondary
terminal
voltage is 2200 volts and the transformer is called a step-up
,
r
Ns
Np
where Vs and Vp are the secondary and primary terminal
voltages and Ns and Np are the number of turns in the
secondary and primary windings respectively.
If there are 20 turns in the secondary winding for every
turn of the primary, the transformer is said to have a turns
,
,
521
INDUCTION
r'
transformer. If the connections are reversed and the coil
with the larger number of turns is made the primary, the
transformer becomes a step~down transformer. The same
primary voltage now produces a voltage across the secondary of 5.5 volts.
It can be shown that when power is delivered to a load
in the secondary circuit, the product of the secondary current and secondary turns is essentially equal to the product of the primary current and primary turns.
IsNs
or
Is
-
Ip
=
IpNp
-
Np
Ns
Thus when the voltage is stepped up, the current is
stepped down; there is no power gain as a result of transfonner action. Ideally primary and secondary power are
equal, but actually there are power losses as in any machine.
The efficiencies of practical transformers are high and
constant over a wide power range. Transformer efficiencies above 95% are common. Efficiency can be expressed
as the ratio of the power dissipated in the secondary circujt to the power used in the primary.
p
Efficiency = ~ X 100%
Pp
20.22 Transformer
Losses
While transformer efficiencies are high, the transfer of energy from the primary circuit to the secondary circuit does not occur without some
Figure 20-37. An external view,
low-voltage side of a power transformer.
522
Recall that resistance varies in~
versely with cross-secUonal area.
CHAPTER 20
loss. We shall consider two types: copper losses and eddy~
current losses. They represent wasted energy and appear as
heat.
1. Copper losses. These losses result from the resistance
of the copper wires in the primary and secondary turns.
Copper losses are [2R heat losses. They cannot be
avoided.
2. Eddy-currellt losses. When a mass of conducting metal
is moved in a magnetic field or is subjected to a changing
magnetic flux, induced currents circulate in the mass.
These closed loops of induced current circulating in planes
perpendicular to-the magnetic flux are known as eddy currents.
Eddy currents in motor and generator armatures and
transformer cores produce heat due to the [2R losses in the
resistance of the iron. They are induced currents that do
no useful work, and they waste energy by opposing the
change that induces them according to Lenz's law. Eddycurrent losses are reduced by laminating the armature
frames and cores. Thin sheets of metal with insulated surfaces are used to build up the armatures and cores. The
laminations are set in planes parallel to the magnetic flux
so that the eddy-current loops are confined to the width of
the individual laminations. The high resistance associated
with the narrow width of the individual laminations effectively reduces the induced currents and thus the ]2R heating losses.
QUESTIONS:
GROUP A
1. What is mutual inductance?
2. Differentiate between the primary coil
and the secondary coil.
3. (a) To what quantity in mechanics is
inductance analogous? (b) vVhy?
4. Describe the principal parts of a
transformer.
5. What is it that a transformer "transforms"?
6. What is the difference between a
"step-up" and "step-down"
transfonner?
7. (a) Does a "step-up" transformer step
up power? (b) Energy? (c) What law
governs these transformations?
GROUP B
8. A calculator runs On a 9-V battery but
has an adapter so you can plug it into
a wall outlet and run it off 1l0~ V al~
ternating current. What are the functions of this "adapter"?
9. The power produced at a generating
station is of relatively low voltage
(several thousand volts), as is the
power needed in homes (llO-V).
(a) What would happen if this lowvoltage, high~current power were sent
out over transmission lines? (b) How
is this avoided?
10. A physics demonstration
consists of
dropping a bar magnet down a copper pipe. If the pipe is of sufficient
length, it takes an appreciably longer
time for the magnet to fall than if it
had been dropped from the same
height outside the tube. Why?
"
'\
-
I'
1<
52.1
ELECTROMAGNETIC INDUCTION
PROBLEMS:
GROUP A
1. A pair of adjacent coib has a mutual
inductance of 1.06 H. Determine the
average emf induced in the secondary
circuit when the current in the primary
circuit changes from 0.00 A to 9.50 A
in 0.033 6 s.
2. When the primary circuit of a pair of
adjacent coils is activated, the current
surges to 12 A in 0.048 s and the emf
induced in the secondary circuit is
270 V. Determine the mutual inductance of the pair of inductors.
3. A step-up transformer is used on a
120-V line to provide a potential difference of 2400 V. If the primary has 75
turns, how many turns must the sec~
ondary have (neglecting losses)?
4. An initial rise in current in a coil occurs at the rate of 7.5 A/s at the instant
a potential difference of 16.5 V is applied across it. (a) What is the selfinductance of the coil? (b) At the same
instant a potential difference of 50 V is
induced across an adjacent coil. Find
the mutual inductance of the two coils.
5. A coil with an inductance of 0.42 H
and a resistance of 25 n is connected
across a 1l0-V d-c line. What is the
rate of current rise (a) at the instant
the switch is closed; (b) at the instant
SUMMARY..
...
...
6.
7.
8.
9.
the current reaches 85% of its steadystate value?
AS: 1 step-down transformer with
negligible losses is connected to a
120-V a-c source. The secondary circuit
has a resistance of 15.0 n. (a) What is
the potential difference across the secondary? (b) What is the secondary current? (c) How much power is dissipated in the secondary resistance?
(d) What is the primary current?
Assume that the transformer of Problem 6 is replaced by one having an efficiency of 92.5%. What is the primary
current?
A transformer with a primary _of 400
turns is connected across a 120-Va-c
line. The secondary ci~uit has a potential difference of 3000 V. The secondary current is 60.0 mA and the primary current is 1.85 A. (a) How many
turns are in the secondary winding?
(b) What is the transformer efficiency?
A 270-H resistor, a 2.50-H coil, and a
switch are connected in series across a
12.0-V battery. At a certain instant
after the switch is closed the current is
20.0 mA. What is the potential differw
ence (a) across the resistor; (b) across
the inductor? (c) What is the rate of
change of current at this instant?
mil.
An emf is induced in a conductor when
relative motion between the conductor
and a magnetic field produces a change
in the flux linkage. The greater the rate of
relative motion, the greater is the magnitude of the induced emf. If the conductor
is part of a closed circuit, an electron current is induced in the circuit. The direction of an induced current is always in
accord with Lenz's law.
An electric generator converts mechanical energy into electric energy. The direc-
hon of induced electron current in the
armature turns is determined by use of
the left-hand generator rule. The generator induces a sinusoidal emf across the
armature turns. The current in the armature circuit alternates. The frequency of
the generated current is expressed in
hertz; one hertz is equivalent to one cycle
per second. An a-c generator may be
modified for a pulsating d-c output. The
d-c generator is self-excited.
Electric motors convert electric energy
524
CHAPTER 20
into mechanical energy. The motor effect
is the result of an electric current in a
magnetic field; it is the reverse of the
generator effect. The direction of motion
of the armature turns is determined by
the use of the right-hand motor rule. An
electric motor produces a back emf that
subtracts from the applied voltage. Practical d-c motors are of three types: series,
shunt, and compound wound. Three
common types of a-( motors are the uni~
versal motor, the induction motor, and
the synchronous motor. Of these, the
induction motor is most widely used.
If a change in current in one circuit
induces an emf in a second circuit, the
two have a property of mutual inductance. An emf is induced across a coil by
a change of current in the coil; this property is known as self-inductance.
The unit
of inductance is the henry. The induced
emf across a given inductance depends
on the time rate of change of current in
the inductance. The property of inductance is described as a kind of electric
inertia.
Transformers are alternating-current
devices. Primary and secondary windings
have a common core. For a given primary
voltage, the turns ratio determines the
secondary voltage. The transformer may
either step up or step down the a-c voltage of the primary circuit. Efficient and
economical distribution of a-c power is
possible through the use of the transformer principle.
VOCABULARY
alternating current
back emf
commutator
compound-wound
generator
eddy currents
electromagnetic induction
excitor
force couple
generator rule
induced current
induced emf
instantaneous
current
instantaneous
voltage
Lenz's law
magnetic force
magneto
motor effect
motor rule
mutual inductance
rotor
self-inductance
series-wound generator
shunt-wound
generator
slip ring
stator
synchronous motor
three-phase generator